By Yasuo Narukawa

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**Example text**

Let S = {x1 , . . , xn } be a sample, and let Tn be an estimator; then, the breakdown point of the estimator T at the sample S is deﬁned as 1 ∗ min{m; bias(m; T, S) = ∞}, n (T, S) := n with bias(m; T, S) deﬁned by bias(m; T, S) := sup S ∈R(S,m) ||T (S ) − T (S)||, where R(S, m) represents all samples obtained from S with m original observations replaced by arbitrary values. Note that, here, bias(m; T, S) < ∞ means that the eﬀect of m perturbations is bounded, while bias(m; T, S) = ∞ means that it is not.

It is well known that the higher P (A), the higher the likelihood that A occurs. As events are subsets of X, probability measures are set functions. When ﬁnite sets X are considered, probability measures can be deﬁned on all subsets of X. That is, P is a function on the set ℘(X) into [0, 1]. Nevertheless, in general, it is not possible to consider all subsets of X. This is the case, for example, when X is not ﬁnite. In such situation, measures are deﬁned over σ-algebras. They are subsets A of ℘(X) with some particular properties.

Temperatures are an example of interval scales. 8C + 32. In interval scales, the ratios of intervals are invariant. This is formally expressed by ψ(φ(a1 )) − ψ(φ(a2 )) φ(a1 ) − φ(a2 ) = . φ(b1 ) − φ(b2 ) ψ(φ(b1 )) − ψ(φ(b2 )) 4. 5. Any monotone increasing function is a permissible transformation. The Mohs scale of hardness is an example of this scale. Preferences are also often expressed using ordinal scales. Any ordered set of values is equally appropriate to express the ordering. In the case of preferences, it is equally valid to use the set {1, 2, 3, 4, 5}, the set {A, B, C, D, E}, or the set {very low, low, medium, large, very large} to express which alternative we prefer.